Since this is a suggestion that comes up quite often, I figured I'd share an example to illustrate how this is a bit trickier than it seems.
In this screenshot, the mobile chat is semi-transparent so that you can see the game world underneath. This presents the following issues:
The text is a lot harder to read, especially on smaller screens
The HUD buttons are jumbled together with the chat buttons, adding to confusion
You can't actually interact with the game world underneath the chat so it somewhat defeats the purpose
While it might seem like it could fill perhaps only half the screen, the issue is one of screen space - the above is optimized for medium-sized tablets and on an average phone, chat is already a bit too small as it is.
Some of you have suggested just making the HUD popup bubble semi-transparent, but this would present the same issue of text legibility and would not add much functionality.
In light of these constraints and given that our chat is much more advanced than just a single-channel chatroom as in some similar games, we have instead chosen to acknowledge the limitations of mobile devices, envision chat as something you choose to enter and engage with, and simply make that the best experience it can be. (note that I'm not talking about the HUD bubbles here, which I agree can be made better by filtering out channels)
While PC will always be more convenient to chat in due to the vastly larger screens, hopefully this can illustrate why the distinction exists.
There's actually another reason the chat is not transparent too, and it's the same reason the other game windows also aren't transparent: While any window on mobile is open, the game actually stops rendering the game world in the background. This saves on battery and heat, and makes the UI feel much smoother to use since there's no need to draw anything behind the window. While we could certainly live without this feature, it's quite a useful one!